Study on the Dynamic Response of the Component Failure of Drum-Shaped Honeycomb-Type III Cable Dome with Quad-Strut Layout

Abstract

The drum-shaped honeycomb-type cable dome departs from traditional concepts and incorporates the idea of multiple strut configurations. It is the most diverse type of cable dome structure. Both cables and struts serve as the main load-bearing components. Analyzing the effects of local component failure is crucial for designing large-scale cable domes able to resist continuous collapse. A numerical analysis model using the ANSYS finite element method with a 120 m span is established. Dynamic analysis methods are employed to study the response of structural internal forces and displacements during the failure of different component types. By defining the internal force dynamic coefficient and change coefficient, the structural continuity collapse resulting from component failure is determined based on computational results and observation of structural deformations. Furthermore, the importance of the various components within the overall structure is classified. The research findings indicate that the failure of individual components does not cause overall instability of the structure. The importance level of the ring cables and some upper chord ridge cables is higher than that of the inclined cables. Class a struts have a higher importance level than class b struts. Additionally, the importance level of outer ring components is higher than that of inner ring components.

1. Introduction

The cable dome theory originated from the concept of “tensegrity” proposed by the architectural master R.B. Fuller [1]. In 1986, the renowned American engineer Geiger, based on this concept, invented a prestressed tensegrity cable dome structure supported by a peripheral compressive ring beam, known as the Geiger-type cable dome structure. In 1992, American engineer Levy made improvements to it and proposed the Levy-type cable dome. However, both of these classical cable dome structures have their own drawbacks. The Geiger-type cable dome [2] has weak out-of-plane stiffness at the nodes on each strut, making it prone to instability. The Levy-type cable dome [3] solves the instability problem of the Geiger-type cable dome, but increases the number of components due to the arrangement of the structure. Additionally, the dense subdivision of the inner grid makes the construction of the inner nodes and membrane laying more difficult.

To address the issues with these two classic cable dome structures, in 2005, Dong Shilin et al. [4] proposed the Kiewit-type cable dome and two hybrid dome structures, considering geometric topology, structural construction, and force mechanics. They applied various grid shell configurations to cable dome structures, all of which have the advantages of uniform grid division and even stiffness distribution. In 2010, Dong Shilin et al. [5] creatively introduced a spatial structural system that combines a single-layer grid shell with a cable dome. Subsequently, they proposed the honeycomb-type cable dome with a multi-strut layout [6], where the upper chord of the dome is composed of ridge cables distributed in a honeycomb pattern and connected to the lower chord by multiple struts. This arrangement departs from the traditional concept of the tensioned structure. Following this innovative research on the drum-shaped honeycomb sequential cable dome structural system, Dong Shilin recently proposed a drum honeycomb-type cable dome with a quad-strut layout [7] and explored analysis methods for the prestressed states of this cable dome. The four compression struts are connected as a whole at the lower chord node, significantly improving the stability of the structure compared to the Geiger-type cable dome. The upper chord ridge cables have three schemes: Type I, II and III. Type III consists of a plum blossom arrangement comprisig a drum honeycomb hexagonal main and sub-grid system, which can also be seen as a circumferentially alternating arrangement of drum honeycomb hexagonal grids offset by half a grid in the radial direction.

In large cable structures, the tension cables and compression struts are the main load-bearing members, and bear relatively large loads, while damage to local members may occur under the action of chance events or when the structure is severely overloaded, leading to sudden changes in the geometry of the structure and vibration because the internal forces of the structure are no longer in equilibrium, until the internal forces of the structure reach equilibrium again or damage failure. In recent years, a series of studies and comparisons have been carried out by domestic and foreign scholars using different analysis methods related to local member failure and continuity collapse resistance of pretensioned structures with cables. Ying Yu et al. [8] employed the finite particle method to perform verification and anti-collapse analysis on cable net structures under strong wind action, proposing methods to address geometric nonlinearity and material nonlinearity in discrete elements. Xiaoxi Wang et al. [9] investigated the deformation and internal force redistribution process of a scaled-down model of a 10.8 m chord-supported cable dome after cable failure through numerical simulation and experiments. The results indicate that cable failure leads to significant oscillation effects in the remaining structural components. Renjie Liu et al. [10] conducted an analysis of the anti-continuous collapse of Levy-type geodesic domes and loop-free suspen-dome after cable failure using the AP method. They discussed the structural displacement and load-carrying capacity after implementing 34 different cable failure scenarios. Chao Zhang et al. [11] utilized an improved alternative load path implicit analysis method to simulate the beam-cable failure in a suspen-dome. They analyzed parameters including the failure time, failure path, and damping ratio, thereby providing recommended numerical values. Zhenyu Xu et al. [12] proposed the mechanism of continuous collapse for a suspen-dome structure under the cable fracture based on experimental and finite element results. Kahla et al. [13] carried out a failure analysis of cables in tensioned monolithic structures. Ran Zhao et al. [14] studied the analysis of cable breakage of the cable membrane structure of Baoan Stadium in Shenzhen and gave the static response of the structure under constant load. Lianmeng Chen et al. [15] conducted a cable rod breakage analysis for a rib-ring type cable dome, but the dynamic time response process was not given. Jinyu Lu et al. [16] and Xiaofeng Jiang et al. [17] analyzed the dynamic response and collapse process of the structure after the breakage of the cable-rod tension structure and tension string beam. Zhang Chao et al. [18] analyzed the local breakage of a multiple-quadrilateral ring cable-tensioned chord dome and gave the dynamic amplification factor of the chord dome. Zhu et al. [19] analyzed the structural dynamic response of the chord-supported dome structure after damage to the cable and strut. Haoqing Liang et al. [20] analyzed the effect of local cable rod failure on the structural performance of rib ring herringbone cable dome, and gave the dynamic time response process. Chenyu Liang, Zhongyi Zhu et al. [21] selected several large domestic and foreign cable structures, including the FAST telescope, the National Speed Skating Stadium, and Qatar Lusail Stadium, and analyzed and compared the dynamic response characteristics and dynamic coefficients of different types of cable structure, such as the cable net and spoke-type cable trusses with hair fracture index.

Accordingly, in order to further study the changes in structural force performance of the drum-shaped honeycomb type III cable dome with a quad-strut layout structure when the local cable rod breaks, in this paper, we establish an ANSYS finite element numerical model. Nonlinear dynamic analysis methods are employed to study the dynamic response of structural internal forces and displacements during the failure of various cable components. Based on the computational results and by defining force dynamic coefficients, force change coefficients, and observing the structural configuration after stability restoration, the importance of corresponding components within the overall structure is determined. This research provides a reference for the practical engineering design of this cable dome configuration.

2. Structural Model and Calculation Method

2.1. Structural Model

A numerical model was built of a structure with a span of 120 m and an inner circular opening diameter of 40 m. The structural model is shown in Figure 1. Previous research suggests that the structure’s prestress distribution and stiffness are more optimal when the structural rise–span ratio and height–span ratio are 0.08 and 0.10, respectively [7]. The structure consists of 17 different types of component, including 13 tension cables and 4 compression struts. For modeling these components, the Link180 element was chosen due to its ability to handle large strains and deformations. Additionally, the element allows for the specification of whether the struts experience tension or compression, making it suitable for simulating the elements in this structural model. The cable sections are determined based on a prestress level of 20% of the cable’s breaking force. The sizing of the strut sections follows the design principle of maintaining a suitable aspect ratio, ensuring comparable initial prestress among all components. The cables are modeled as tension elements, while the struts are modeled as compression elements, considering the effects of large deformations and stress stiffening. In the practical application of the structure, the outermost nodes of the cable dome structure are connected to a rigid ring beam. Therefore, all nodes on the outer ring of the structure are constrained and fully fixed in the APDL command flow, applying all constraints. In addition to considering the self-weight of the structure, a roof load of 0.5 kN/m² is applied as the equivalent concentrated load at the bi-directional guided nodes. The initial prestress of the structure is obtained using the singular value decomposition method [22], and is applied by imposing initial strains. The initial prestress values and component section parameters are detailed in

Table 1. Table of structural prestressing modal and member section parameters.

Component	Prestressed Mode	Self-Equilibrating Internal Force (kN)	Section Dimension
N1	0.52	5235	Φ147
N1a	0.60	6030	Φ161
N1b	0.18	1760	Φ92
N2a	0.40	3962	Φ127
N2b	0.54	5448	Φ147
T1a	0.23	2322	Φ80
T1b	0.27	2660	Φ107
T2a	0.49	4879	Φ147
T2b	0.60	5980	Φ155
B1	0.11	1108	Φ92
B2	0.47	4699	Φ147
H1	0.22	2240	Φ80
H2	1.00	10,000	Φ154
V1a	−0.03	−294	Φ254 × 14
V1b	−0.02	−231	Φ273 × 16
V2a	−0.11	−1082	Φ426 × 16
V2b	−0.08	−789	Φ427 × 15

. Material parameters are shown in

Table 2. Material parameters.

Components	Yield Strength (Mpa)	Poisson’s Ratio	Modulus of Elasticity (Mpa)	Coefficient of Linear Expansion	Density (kg/mm3)
cables	1860	0.3	1.95 × 105	1.36 × 10−5	7.85 × 10−6
struts	345	0.3	2.06 × 105	1.2 × 10−5	7.85 × 10−6

. As shown in Table 1, after the finite element structural model self-equilibrates, the internal force distribution is aligned with the theoretical calculation results, confirming the correctness and appropriateness of the established numerical model for the structure.

2.2. Dynamic Analysis Methods

The basic principle of the power analysis method for failure detection in linkages is structural dynamics, with its structural dynamic equation given as follows:

$$[M]\left\{\ddot{u}\right\}+[C]\left\{\dot{u}\right\}+[K]\left\{u\right\}=\{F(\mathrm{t})\}$$

(1)

In the aforementioned equation, [M] represents the structural mass matrix, [C] represents the structural damping matrix, [K] represents the structural stiffness matrix, {F(t)} denotes the time-varying load function, $\left\{u\right\}$ is the nodal displacement vector, $\left\{\dot{u}\right\}$ is the nodal velocity vector, and $\left\{\ddot{u}\right\}$ is the nodal acceleration vector [23]. The solution of the aforementioned motion equation can be obtained through two main methods: the modal superposition method and the direct integration method. The modal superposition method transforms the coupled motion equations, based on natural frequencies and mode shapes, into a set of independent uncoupled equations. On the other hand, the direct integration method solves the motion equations directly through time integration and can be implemented using either explicit or implicit approaches.

Transient dynamic analysis in the general finite element analysis software ANSYS is used to determine the dynamic response of structures under arbitrary time-varying loads, such as impact loads and sudden loads. It includes three methods: the direct method, the reduced method, and the modal superposition method. Among them, the direct method calculates the transient response using the complete system matrix and is the most powerful method among the three, capable of including various nonlinear characteristics (plasticity, large deformation, and large strain). ANSYS provides the functionality of the element birth and death technique, making it possible to simulate practical engineering problems such as excavation, structure installation and removal, and damage by adding and removing selected elements. In ANSYS, killed elements are only multiplied by a very small factor (1 × 10⁻⁶) [24] of their stiffness and are not actually removed from the model. When using the element birth and death technique, killing elements may result in individual nodes that are not connected to any surviving elements, causing these nodes to “drift”, eventually leading to non-convergence of the calculations. In this case, it is necessary to constrain these potentially drifting nodes to reduce the number of equations to be solved and prevent non-convergence. The entire dynamic process of component failure is calculated using ANSYS transient dynamic analysis (direct method). The element birth and death technique is employed to simulate component failure, considering large deformations for nonlinear dynamic analysis. The damping ratio of the cable dome structure is taken as 0.02, and the duration of component unloading is set to 0.001 s. It is assumed that the cable failure occurs at the end of the first second, and the total duration of the dynamic analysis is not less than 8 s. The unloading process of the component is shown in Figure 2, where F₀ represents the initial internal forces of the structure.

2.3. Component Numbering and Definition of Relevant Coefficient

Figure 3a illustrates the highly symmetrical configuration of the drum honeycomb quadrilateral truss Type III dome. It can be regarded as a substructure formed by rotation along the central axis. To facilitate calculation and analysis, the structure is divided into five substructures based on the radial symmetric axis. Each substructure undergoes controlled failure of different categories of cable and strut elements in the first substructure, while monitoring the changes in internal forces of the remaining four substructures’ components. The component labeling, as depicted in Figure 3b, assigns identifiers N, T, B, H, and V to represent ridge cables, diagonal cables, ring cables, and struts, respectively. The numerical subscripts indicate the concentric circles from the innermost to the outermost, while the alphabetical subscripts represent the categories of the components. The numerical values (1–5) following the component names indicate the substructure number to which the component belongs. For instance, N_1a-1 denotes an upper chord ring cable of category ‘a’ in the first substructure.

In the subsequent table documenting the variations in internal forces, F represents the internal force of the components before failure, F_max represents the maximum internal force among the components after failure, F_min represents the minimum internal force among the components after failure, and F_dmax indicates the maximum value of internal force fluctuations during the post-failure period. The internal force dynamic coefficient β₁ and the internal force variation coefficient β₂ are defined to evaluate the changes in internal forces of the members. If β₁ > 1.5 or β₂ > 0.5, it is considered that the internal force variations of that particular member are significantly influenced by the failure of the component. The specific formulas are as follows:

$${\beta }_1=\frac{F_{\mathrm{dmax}}}{F}$$

(2)

$${\beta }_2=\frac{(F_{\max }-F_{\min })}{F}$$

(3)

2.4. Rules for Differentiating Component Importance

This study comprehensively assesses the importance of failed components in the overall structure based on the computational results of structural dynamic responses after component failure, including the internal force dynamic and variation coefficients, the number of relaxation components, and the maximum displacement of the structure. The importance level of the components will be distinguished according to the following rules:

• After a single component breaks, it causes structural deformation. U_Zmax represents the maximum vertical displacement after structural stability is restored. The “+” and “−” represent displacement in the positive or negative directions along the Z-axis. If its absolute value is greater than 0.48 (Span/250), it is considered that the structure has experienced local failure, and the importance level of the broken component is at least medium.
• After a single component breaks, the number of significantly affected components is determined based on the value of the β₁ and β₂. Simultaneously, the changes in internal forces of the remaining components are tracked. After the structure restores stability, the number of relaxation components is determined based on the presence of zero internal forces in the cables. If the number of significantly affected components is greater than 0 or the number of relaxation components is greater than 0, the importance level of the broken component is considered to be medium. If both the number are greater than 0, the importance level of the broken component is considered to be high.
• After a single component breaks, if the structure does not experience local failure and there are no significantly affected components or relaxation components, it can be considered that such components have a low level of importance in the structure.

3. The Impact of Local Component Failure on the Structural Load Bearing Performance

3.1. Failure of Component N₁-1

After the failure of the component N₁-1, the time history curves of the internal forces and vertical displacements of ridge cable N₁ are presented in Figure 4a,b. Analysis of the graph reveals that following the rupture of N₁-1 in the upper chord ring cable, the remaining cables of the same category exhibit relaxation behavior, with their internal forces rapidly decreasing to lower levels. The vertical displacements of the nodes show fluctuations towards the end of the first second, characterized by three distinct oscillations before returning to a stable state. Notably, the vertical displacements of upper chord nodes 1_a, 1_b, and lower chord node 1′ undergo significant changes, while the other nodes remain relatively stable.

Table 3. Variation of internal forces in various components after the failure of N₁-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4830.3	75.6	71.8	1.00	0.01
N1a	5596.6	5410.4	95.5	72.6	1.00	0.01
N1b	1620.9	5590.1	58.1	5.65	3.45	0.03
N2a	3771.1	3750.8	176.4	157	1.00	0.01
N2b	5150.2	5100.2	200.1	185	1.00	0.01
T1a	2154.1	2400.8	69.6	10.5	1.11	0.03
T1b	2452.6	2590.3	28.7	10.3	1.06	0.01
T2a	4651.5	3280.1	227.8	199.1	0.71	0.01
T2b	5661.3	4750.5	230.5	209.3	0.84	0.00
B1	1143.2	1950.4	163.3	144.3	1.71	0.02
B2	4987.7	4990.8	1110.6	634.6	1.00	0.10
H1	2318.9	3820.1	331.6	318.4	1.65	0.01
H2	10,668.1	10,900.2	2020.5	1770.7	1.02	0.02
V1a	−287.9	−390.5	−26.8	−4.6	1.35	0.08
V1b	−231.0	−1120.6	−33.5	−8.9	4.85	0.11
V2a	−1113.5	−1360.3	−155	−133.2	1.22	0.02
V2b	−787.1	−788.1	−82.9	−65.1	1.00	0.02

indicates that the failure of the component N₁-1 has a significant impact on ridge cable N_1b and the inner circle strut V_1b of the lower chord. The internal force coefficient β₁ of strut V_1b reaches 4.85, and the internal force variation coefficient β₂ reaches 0.11, whereas the internal force coefficients for other components are relatively small. Additionally, Figure 4c illustrates the structural deformation at 9 s after failure. It shows that, following the stabilization of structural deformations, a localized collapse occurs at the position of the failed component N₁-1, while the overall structure remains intact without experiencing complete failure.

3.2. Failure of Component N_1a-1

After the upper chord ridge cable N_1a-1 failed, the force and vertical displacement time history curves of spinal cord N_1a are depicted in Figure 5a,b. The internal force of spinal cord N_1a experiences a significant reduction, and there is a noticeable fluctuation in the vertical displacement of the lower node 1’ at the end of the first second. However, the displacement changes in other nodes are relatively minor.

Table 4. Variation of internal forces in various components after the failure of N_1a-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4856.9	86.3	43.3	1.00	0.01
N1a	5596.6	5596.6	101.4	102.9	1.00	0.01
N1b	1620.9	3847.6	22.8	0.0	2.37	0.02
N2a	3771.1	3771.5	177.5	160.9	1.00	0.01
N2b	5150.2	5149.4	205.7	189.4	1.00	0.01
T1a	2154.1	2154.1	37.9	14.1	1.00	0.01
T1b	2452.6	2452.6	31.5	0.0	1.00	0.02
T2a	4651.5	4651.5	229.1	195.9	1.00	0.01
T2b	5661.3	5659.6	234.9	208.4	1.00	0.01
B1	1143.2	1729.6	158.8	126.7	1.51	0.03
B2	4987.7	4991.0	881.1	633.8	1.00	0.05
H1	2318.9	3797.6	338.5	318.2	1.64	0.01
H2	10,668	10,775.9	1959.5	1798.2	1.01	0.02
V1a	−287.9	−287.9	−20.6	−9.3	1.00	0.04
V1b	−231.0	−565.0	−25.2	−11.8	2.45	0.06
V2a	−1114	−1318.9	−150.0	−134.1	1.18	0.01
V2b	−787.1	−787.2	−82.4	−60.0	1.00	0.03

reveals that spinal cords N_1b and T1b exhibit relaxation. Notably, the failure of N_1a-1 has the most pronounced impact on the internal brace V_1b of the inner ring. Following the redistribution of internal forces within the structure, the internal force dynamic coefficient β₁ for brace V_1b reaches 2.45, and the internal force variation coefficient β₂ for internal forces reaches 0.06, while the coefficients for other components remain comparatively small. From Figure 5c, it is evident that after the structural deformation stabilizes, the location of the failed spinal cord N_1a-1 experiences localized collapse, but overall structural failure does not occur.

3.3. Failure of Component N_1b-1

After the failure of the upper chord ridge cable N_1b-1, the force and vertical displacement time history curves of spinal cord N_1b are depicted in Figure 6a,b. Following a notable fluctuation, the structure once again reaches an equilibrium state. N_1b, which connects the upper node 1_b of two adjacent substructures, exhibits a distinct fluctuation in internal forces at the end of the first second. While the internal forces of spinal cord N_1b decrease, there is no indication of relaxation. Simultaneously, the vertical displacement of node 1_b rapidly increases to 2 m, while the displacement of other nodes remains relatively minimal. The changes in internal forces for other components are presented in

Table 5. Variation of internal forces in various components after the failure of N_1b-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	6567.4	4123.5	3251.0	1.35	0.18
N1a	5596.6	6399.3	4086.3	3927.7	1.14	0.03
N1b	1620.9	1945.7	1375.8	1094.3	1.20	0.17
N2a	3771.1	4019.7	2805.1	2481.7	1.07	0.09
N2b	5150.2	5973.2	4120.5	3475.3	1.16	0.13
T1a	2154.1	2659.9	2052.1	1018.2	1.23	0.48
T1b	2452.6	3009.2	2246.2	817.6	1.23	0.58
T2a	4651.5	5078.1	3638.4	2919.1	1.09	0.15
T2b	5661.3	6294.2	4420.1	3761.5	1.11	0.12
B1	1143.2	2426.4	1993.1	3.9	2.12	1.74
B2	4987.7	5088.8	4179.8	3426.3	1.02	0.15
H1	2318.9	2962.1	2054.9	1471.0	1.28	0.25
H2	10,668.1	11,328.1	8469.7	8076.1	1.06	0.04
V1a	−287.9	−318.5	−209.3	−72.2	1.11	0.48
V1b	−231.0	−749.5	−399.1	−96.1	3.24	1.31
V2a	−1113.5	−1754.9	−1481.5	−241.3	1.58	1.11
V2b	−787.1	−788.1	−632.5	−363.8	1.00	0.34

. The failure of spinal cord N_1b-1 has a considerable impact on B₁ and V_1b. After the redistribution of internal forces within the structure, the diagonal cable B₁ experiences relaxation, and the internal force dynamic coefficient β₁ for brace V_1b reaches 3.24, while the internal force variation coefficient β₂ for internal forces reaches 1.31. As illustrated in Figure 6c, after the rupture of N_1b-1, the maximum deformation occurs at node 1_b, signifying localized structural failure without continuous collapse. Nevertheless, the structure retains its load-bearing capacity.

3.4. Failure of Component N_2a-1

Figure 7a,b illustrates the time history curves of internal forces and vertical displacements of nodes after the failure of the circumferential spinal cord N_2a-1. From Figure 7a, it is evident that the internal forces in N_2a rapidly decrease to a significantly lower level. The displacements of various nodes in the structure exhibit considerable variations, leading to uneven deformations. Both the upper node 2_a and the lower node 1′ experience substantial deformations, with the maximum displacement reaching 4 m after stabilization.

Table 6. Variation of internal forces in various components after the failure of N_2a-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4855.8	277.5	240.0	1.00	0.01
N1a	5596.6	5596.6	324.0	286.1	1.00	0.01
N1b	1620.9	1728.4	79.8	0.0	1.07	0.03
N2a	3771.1	3770.6	326.8	257.0	1.00	0.02
N2b	5150.2	5155.0	400.9	226.8	1.00	0.03
T1a	2154.1	2154.1	129.5	110.6	1.00	0.01
T1b	2452.6	2561.4	128.6	102.5	1.04	0.01
T2a	4651.5	4638.2	406.7	249.7	1.00	0.03
T2b	5661.3	5658.0	448.8	360.3	1.00	0.02
B1	1143.2	2201.6	195.1	135.3	1.93	0.05
B2	4987.7	5031.6	1075.7	0.0	1.01	0.22
H1	2318.9	2551.8	407.2	365.5	1.10	0.02
H2	10,668.1	10,692.7	2373.7	1475.2	1.00	0.08
V1a	−287.9	−287.9	−35.6	−15.8	1.00	0.07
V1b	−231.0	−511.4	−40.8	−28.6	2.21	0.05
V2a	−1113.5	−1302.4	−189.7	−34.6	1.17	0.14
V2b	−787.1	−1291.3	−111.7	−62.1	1.64	0.06

provides insights into the changes in internal forces for other components. It is noticeable that the failure of N_2a-1 has a significant impact on N_1b, B₁, and V_1b. Both the spinal cord N_1b and the outer ring diagonal cable B₂ exhibit relaxation. The internal force dynamic coefficient β₁ for brace V_1b reaches 2.21, while the internal force variation coefficient β₂ for internal forces reaches 0.05. At the 9th second after the failure of N_2a-1, the structural deformation, as depicted in Figure 7c, shows localized failure, but the structure is still capable of bearing the load.

3.5. Failure of Component N_2b-1

Following the failure of the upper chord ridge cable N_2b-1, the time history curves of internal forces and vertical displacements of nodes are presented in Figure 8a,b. The internal forces in spinal cord N_2b exhibit a substantial decrease without any sign of relaxation. The upper node 2_b, connected to N_2b, displays noticeable fluctuations in displacement. Upon reaching a stable state of deformation, the displacement at node 2_b reaches 4.3 m, while the remaining nodes demonstrate relatively stable changes.

Table 7. Variation of internal forces in various components after the failure of N_2b-1.

Components	F(kN)	Fdmax(kN)	Fmax(kN)	Fmin(kN)	β1	β2
N1	4855.9	4855.9	98.5	86.8	1.00	0.00
N1a	5596.6	5595.6	114.8	109.4	1.00	0.00
N1b	1620.9	2636.1	48.6	0.0	1.63	0.03
N2a	3771.1	3777.4	184.8	109.5	1.00	0.02
N2b	5150.2	5162.4	209.6	168.0	1.00	0.01
T1a	2154.1	2154.0	52.9	33.6	1.00	0.01
T1b	2452.6	3923.1	60.0	0.0	1.60	0.03
T2a	4651.5	4643.9	231.5	186.7	1.00	0.01
T2b	5661.3	5641.7	237.2	129.4	1.00	0.02
B1	1143.2	2012.8	151.8	124.1	1.76	0.02
B2	4987.7	5226.4	1031.7	541.4	1.05	0.10
H1	2318.9	2319.3	339.6	291.0	1.00	0.02
H2	10,668.1	11,058.1	1979.6	1613.3	1.04	0.03
V1a	−287.9	−338.7	−24.5	−19.9	1.18	0.02
V1b	−231.0	−433.4	−29.9	−11.4	1.88	0.08
V2a	−1113.5	−1254.0	−150.3	−94.5	1.13	0.05
V2b	−787.1	−2751.5	−146.5	−57.9	3.50	0.11

outlines the variations in internal forces for other components after redistributing the forces. The failure of N_2b-1 notably impacts N_1b, T_1b, and V_2b. Both spinal cord N_1b and T_1b experience relaxation, while the inner ring brace V_1b attains an internal force dynamic coefficient β₁ of 1.88 and an internal force variation coefficient β₂ of 0.05. After the structure stabilizes, the configuration, as depicted in Figure 8c, indicates significant deformation at the upper node 2_b, illustrating localized structural deformations without widespread failure.

3.6. Failure of Component T_1a-1

Following the rupture of upper chord ridge cable T_1a-1, the time-dependent variations of internal forces and vertical displacements of nodes are depicted in Figure 9a,b. The corresponding changes in internal forces for each component are summarized in

Table 8. Variation of internal forces in various components after the failure of T_1a-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	5470.6	4136.4	3645.2	1.13	0.10
N1a	5596.6	5877.5	4374.1	4200.1	1.05	0.03
N1b	1620.9	1805.5	1361.9	794.1	1.11	0.35
N2a	3771.1	3840.8	2949.7	2829.5	1.02	0.03
N2b	5150.2	5415.8	4022.2	3896.6	1.05	0.02
T1a	2154.1	2553.7	1967.5	1625.7	1.19	0.16
T1b	2452.6	2770.6	2156.0	811.1	1.13	0.55
T2a	4651.5	4769.4	3603.1	3317.3	1.03	0.06
T2b	5661.3	5910.9	4378.7	4219.6	1.04	0.03
B1	1143.2	2340.4	1671.2	488.4	2.05	1.03
B2	4987.7	5071.8	4017.5	3771.6	1.02	0.05
H1	2318.9	3021.0	1901.1	1550.2	1.30	0.15
H2	10,668.1	11,105.3	8795.9	8607.3	1.04	0.02
V1a	−287.9	−764.8	−371.4	−214.7	2.66	0.54
V1b	−231.0	−436.0	−445.1	−187.7	1.89	2.74
V2a	−1113.5	−1987.3	−1591.7	−592.7	1.78	0.90
V2b	−787.1	−797.0	−620.2	−380.7	1.01	0.30

. As observed from Figure 9a,b, both the internal forces and node displacements of T_1a exhibit a significant fluctuation before returning to a stable state. The internal forces of all T_1a segments decrease, with the maximum displacement occurring at node 1_b, reaching a magnitude of 3.3 m and stabilizing at 1.9 m at 2 s after rupture. The displacements of other nodes show relatively minor changes. According to Table 8, the rupture of T_1a notably affects B₁, V_1a, and V_1b. The internal force dynamic coefficient β₁ for inner ring brace V_1a reaches 2.66, while internal force the variation coefficient β₂ of internal forces reaches 0.54. As depicted in Figure 9c, after the structural deformation stabilizes, localized failure occurs without resulting in a complete structural collapse.

3.7. Failure of Component T_1b-1

Following the rupture of upper chord ridge cable T_1b-1, the time-dependent variations of internal forces and vertical displacements of nodes are illustrated in Figure 10a,b.

Table 9. Variation of internal forces in various components after the failure of T_1b-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4855.9	98.5	86.8	1.00	0.01
N1a	5596.6	5595.6	114.8	109.4	1.00	0.01
N1b	1620.9	2636.1	98	48.6	1.63	0.03
N2a	3771.1	3777.4	184.8	109.5	1.00	0.02
N2b	5150.2	5162.4	209.6	168.0	1.00	0.01
T1a	2154.1	2154.0	52.9	33.6	1.00	0.01
T1b	2452.6	2734.6	1981.5	1459.6	1.11	0.21
T2a	4651.5	4643.9	231.5	186.7	1.00	0.01
T2b	5661.3	5641.7	237.2	129.4	1.00	0.02
B1	1143.2	2012.8	151.8	124.1	1.76	0.02
B2	4987.7	5226.4	1031.7	541.4	1.05	0.10
H1	2318.9	2319.3	339.6	291.0	1.00	0.02
H2	10,668.1	11,058.1	1979.6	1613.3	1.04	0.03
V1a	−287.9	−338.7	−24.5	−19.9	1.18	0.02
V1b	−231.0	−433.4	−29.9	−11.4	1.88	0.08
V2a	−1113.5	−1254.0	−150.3	−94.5	1.13	0.05
V2b	−787.1	−2751.5	−146.5	−57.9	3.50	0.11

provides an overview of the changes in internal forces for each component. It can be observed from Figure 10a that after experiencing two distinct fluctuations, the internal forces in the structure undergo redistribution. The T_1b internal forces diminish to varying degrees. As the spinal cord T_1b is connected to the upper node 1_b, the vertical displacement at node 1_b exhibits more prominent oscillations, reaching a maximum of 2.4 m. However, the displacements of all nodes stabilize and remain below 1 m. Table 9 reveals that the rupture of T_1b-1 notably impacts N_1b, B₁, and V_1b. Notably, the internal force dynamic coefficient β₁ for inner ring brace V_2b reaches 3.50, while the internal force variation coefficient β₂ of internal forces reaches 0.11. At 9 s after component failure, the structural deformation, as depicted in in Figure 10c, displays deformation at the upper node 1_b, while the overall stiffness remains largely intact.

3.8. Failure of Component T_2a-1

After the rupture of upper chord ridge cable T_2a-1, the variations of internal forces and vertical displacements of nodes over time are depicted in Figure 11a,b.

Table 10. Variation of internal forces in various components after the failure of T_2a-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4858.4	253.0	233.4	1.00	0.01
N1a	5596.6	5595.8	3767.0	3467.6	1.00	0.05
N1b	1620.9	4679.5	80.4	0.0	2.89	0.02
N2a	3771.1	3770.6	308.9	0.0	1.00	0.07
N2b	5150.2	5150.2	368.5	220.8	1.00	0.03
T1a	2154.1	2154.0	106.9	0.0	1.00	0.04
T1b	2452.6	2442.4	119.0	0.0	1.00	0.02
T2a	4651.5	4646.2	424.7	358.3	1.00	0.01
T2b	5661.3	5661.5	472.4	183.7	1.00	0.05
B1	1143.2	2843.9	1997.7	136.9	2.49	1.63
B2	4987.7	5192.4	3707.9	3141.9	1.04	0.11
H1	2318.9	3067.4	1837.4	1401.8	1.32	0.19
H2	10,668.1	11,355.0	7868.0	7439.5	1.06	0.04
V1a	−287.9	−288.0	−35.6	−4.2	1.00	0.11
V1b	−231.0	−856.6	−36.3	−19.5	3.71	0.07
V2a	−1113.5	−1146.9	−202.7	−29.0	1.03	0.16
V2b	−787.1	−971.7	−126.0	−107.9	1.23	0.02

provides a summary of the changes in internal forces for each component. Within the first second following the cable rupture, there is a substantial fluctuation in internal forces, leading to a redistribution of forces within the structure. The T_1b spinal cord experiences a reduction in internal forces to a lower level without relaxation, while the vertical displacement at node 2_b exhibits pronounced oscillations. However, 6 s after the cable rupture, the vertical displacements of all nodes gradually stabilize. According to Table 10, the rupture of T_2a-1 significantly affects the internal forces of components such as N_1b, N_2a, B₁, T_1a, T_1b, and V_1b. Notably, the tension cables N_1b, N_2a, T_1a, and T_1b undergo relaxation. The internal force dynamic coefficient β₁ for the inner ring brace V_1b reaches 3.71, while the internal force variation coefficient β₂ of internal forces reaches 0.07. At 9 s after component failure, the structural deformation, as depicted in Figure 11c, reveals localized significant deformation in the outer ring hexagonal honeycomb grid, but it does not exhibit extensive failure.

3.9. Failure of Component T_2b-1

After the failure of upper chord ridge cable T_2b-1, the variations in internal forces and vertical displacements of each node over time are depicted in Figure 12a,b. T_2b-1, which belongs to the outer ring spinal cord, is connected to nodes 2_b and 3_a. As observed in Figure 12a, when one of these spinal cords of the same type fails, the internal forces of the remaining cords gradually decrease to lower levels. This redistribution of internal forces leads to the relaxation of cables such as N_1b, T_1a, T_1b, and B₁. All nodes experience noticeable vertical displacement fluctuations, with node 2_b exhibiting the largest magnitude, stabilizing at 2.5 m. According to

Table 11. Variation of internal forces in various components after the failure of T_2b-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4856	127.8	116.5	1.00	0.01
N1a	5596.6	5594.41	151.5	136.4	1.00	0.01
N1b	1620.9	1620.92	37.05	0.0	1.00	0.02
N2a	3771.1	3771	211.6	161.4	1.00	0.01
N2b	5150.2	5148	244.3	130.6	1.00	0.02
T1a	2154.1	2153.99	61.5	0.0	1.00	0.02
T1b	2452.6	2456.4	57.2	0.0	1.00	0.02
T2a	4651.5	4651.39	260.3	137.9	1.00	0.03
T2b	5661.3	5660.62	298.2	268.6	1.00	0.01
B1	1143.2	1456.88	184.1	0.0	1.27	0.16
B2	4987.7	4986.36	1185.3	264.7	1.00	0.18
H1	2318.9	2317.94	366.373	323.3	1.00	0.02
H2	10,668.1	10,976	2142.1	1680.7	1.03	0.04
V1a	−287.9	−287.963	−26.9	−15.28	1.00	0.04
V1b	−231.0	−344.435	−27.1	−18.7	1.49	0.04
V2a	−1113.5	−1118.44	−155.4	−71.7	1.00	0.08
V2b	−787.1	−1015.18	−89.3	−13.2	1.29	0.10

, the internal forces of both the inner and outer ring braces of type “b” are significantly affected by the failure of spinal cord T_2b-1. The internal force dynamic coefficient β₁ for the inner ring brace V_1b reaches 1.49, and the internal force variation coefficient β₂ for internal forces reaches 0.04. As shown in Figure 12c, once the structure regains stability, it only undergoes localized deformations, with minimal impact on the overall structure.

3.10. Failure of Component B₁-1

After the failure of the inner ring diagonal cable B₁-1, the internal forces and vertical displacements of each node exhibit time-dependent variations, as shown in Figure 13a,b. B₁ diagonal cable is responsible for providing tension to four adjacent inner ring lower braces, connecting nodes 1′ and 2_a. Upon the failure of B₁-1, there is a noticeable fluctuation in the internal forces of B₁ components within the first second. As the system reaches a new equilibrium, the internal forces of similar cables in neighboring substructures increase, while the remaining components experience varying degrees of reduction. Specifically, the internal force of B₁-2 increases by approximately 20%. Among the vertical displacements of the nodes, node 1_b shows the highest peak at 0.54 m, indicating a notable displacement. However, overall, the displacements remain within acceptable limits, and the structural stiffness is only minimally compromised.

Table 12. Variation of internal forces in various components after the failure of B₁-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	5410.5	4880.5	4710.5	1.11	0.04
N1a	5596.6	6170.2	5580.1	5520.0	1.10	0.01
N1b	1620.9	2070.5	1890.5	1550.5	1.28	0.21
N2a	3771.1	4070.6	3800.9	3720.1	1.08	0.02
N2b	5150.2	5510.1	5190.2	4880.8	1.07	0.06
T1a	2154.1	2390.2	2360.2	2020.8	1.11	0.16
T1b	2452.6	3110.8	2880.2	2310.1	1.27	0.23
T2a	4651.5	5030.9	4770.4	4170.6	1.08	0.13
T2b	5661.3	6110.5	5750.2	5180.8	1.08	0.10
B1	1143.2	1630.2	1400.4	1070.7	1.43	0.29
B2	4987.7	5770.1	5050.6	4440.3	1.16	0.12
H1	2318.9	2690.7	2390.1	2240.5	1.16	0.06
H2	10,668.1	11,800.4	10,700.8	10,300.2	1.11	0.04
V1a	−287.9	−343.1	−304.6	−289.4	1.19	0.05
V1b	−231.0	−270.2	−249.9	−11.1	1.17	1.03
V2a	−1113.5	−1390.5	−1250.2	−685.2	1.25	0.49
V2b	−787.1	−886.1	−855.2	−581.5	1.13	0.36

provides a comprehensive summary of the internal force variations, revealing the impact of B₁-1 failure on various components without significantly affecting any specific component. From Figure 13c illustrating the structural deformation following the component failure, no substantial deformation is observed, affirming the structural integrity of the system.

3.11. Failure of Component B₂-1

After the rupture of inner ring diagonal cable B₂-1, the variation of B₂ internal forces and vertical displacements of each node over time is illustrated in Figure 14a,b. Positioned in the outer ring, B₂-1 acts as the active cable during the initial tensioning phase, imparting equal horizontal tension to all outer ring diagonal cables B₂ to establish the initial prestress in the cable dome structure. Upon the rupture of B₂-1, a noticeable fluctuation in internal forces occurs within 1 s, followed by a redistribution of internal forces throughout the structure. B₂-2 experiences an approximate 26% increase in internal force, while the other similar diagonal cables exhibit varying degrees of reduction in internal forces. Concerning the vertical displacements of each node, the largest oscillation is observed at the upper chord node 2_b of the outer ring, peaking at 0.15 m and gradually stabilizing after the 6^th second. No significant displacements are observed in the remaining nodes.

Table 13. Variation of internal forces in various components after the failure of B₂-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4860.2	4740.7	4710.2	1.00	0.01
N1a	5596.6	5600.4	5460.8	5430.3	1.00	0.01
N1b	1620.9	1660.9	1590.4	1570.1	1.02	0.01
N2a	3771.1	3830.5	3730.6	3640.6	1.02	0.02
N2b	5150.2	5320.1	5050.4	5000.2	1.03	0.01
T1a	2154.1	2190.2	2110.3	2090.7	1.02	0.01
T1b	2452.6	2540.4	2410.2	2370.8	1.04	0.02
T2a	4651.5	4680.6	4550.2	4430.2	1.01	0.03
T2b	5661.3	5670.2	5530.5	5470.8	1.00	0.01
B1	1143.2	1210.5	1150.6	993.4	1.06	0.14
B2	4987.7	7110.1	6310.4	4940.5	1.43	0.27
H1	2318.9	2430.3	2300.1	2250.2	1.05	0.02
H2	10,668.1	12,100.1	11,700.1	10,500.6	1.13	0.11
V1a	−287.9	−341.6	−290.4	−263.8	1.18	0.09
V1b	−231.0	−251.9	−227.1	−216.1	1.09	0.05
V2a	−1113.5	−1160.2	−1120.9	−1020.2	1.04	0.09
V2b	−787.1	−956.2	−786.6	−578.5	1.21	0.26

presents a statistical analysis of the changes in internal forces, revealing that the failure of B₂-1 impacts various components, albeit without significantly affecting any particular component. Furthermore, Figure 14c provides visual evidence that the structure undergoes localized deformation while avoiding total collapse.

3.12. Failure of Component H₁-1

After the rupture of the inner ring cable H₁-1, the internal forces and vertical displacements of each node in H₁ vary over time, as depicted in Figure 15a,b. H₁ is the lower chord ring cable in the inner ring, connecting all lower chord nodes 1′. Following the rupture of H₁-1, the remaining H₁ cables undergo different degrees of oscillation before stabilizing after 5 s. The internal forces in all H₁ cables decrease to varying extents, with the H₁-2 cable closest to the rupture point experiencing a reduction of nearly 50%. Apart from nodes 2_b and 2′, the vertical displacements of other nodes are significantly affected by the cable rupture, gradually converging to stability after 6 s. Notably, node 1_a exhibits the maximum displacement of 2 m.

Table 14. Variation of internal forces in various components after the failure of H₁-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	6010.6	4850.2	4100.2	1.24	0.15
N1a	5596.6	6450.9	5160.2	4730.6	1.15	0.08
N1b	1620.9	2030.5	1580.4	1350.8	1.25	0.14
N2a	3771.1	3830.1	3170.1	3080.6	1.02	0.02
N2b	5150.2	5310.6	4280.4	4020.5	1.03	0.05
T1a	2154.1	3010.2	2430.6	1620.3	1.40	0.38
T1b	2452.6	3150.2	2490.7	2000.9	1.28	0.20
T2a	4651.5	4740.6	3860.9	3480.3	1.02	0.08
T2b	5661.3	5870.8	4690.6	4300.2	1.04	0.07
B1	1143.2	1860.2	1080.7	0.0	1.63	0.94
B2	4987.7	5510.2	4350.2	3330.3	1.10	0.20
H1	2318.9	2890.3	1920.5	1030.4	1.25	0.38
H2	10,668.1	12,100.1	9260.4	8520.3	1.13	0.07
V1a	−287.9	−1180.6	−934.2	−447.1	4.10	1.69
V1b	−231.0	−1400.4	−815.1	59.4	6.06	3.78
V2a	−1113.5	−2180.8	−1320.2	−451.1	1.96	0.78
V2b	−787.1	−933.9	−720.1	−510.6	1.19	0.27

summarizes the changes in internal forces, revealing significant impacts on all bracing members, except V_2b. Specifically, the dynamic internal force coefficient β₁ of the inner ring bracing members V_1a and V_1b reaches values of 4.10 and 6.06, respectively, while the internal force change coefficient β₂ reaches values of 1.69 and 3.68, respectively. By observing the structural deformation Figure 15c, it becomes apparent that a pronounced downward deformation occurs at node 1_a, where the two adjacent substructures connected by H₁-1 are located, indicating a noticeable loss of stiffness in the inner ring structure.

3.13. Failure of Component H₂-1

After the outer ring cable H₂-1 ruptures, the time-varying changes in internal forces of H₁ and the vertical displacements of each node are shown in Figure 16a,b. Acting similarly to H₁, H₂ connects all the lower chord nodes of the outer ring, which constitutes the primary portion of the cable dome structure’s stiffness. Once it breaks, the structural dynamic response becomes severe. Throughout the entire dynamic analysis, H₂ experiences significant fluctuations in internal forces, which subsequently decrease after the redistribution of internal forces within the structure. The vertical displacements of all nodes in the structure exhibit a consistent pattern of change, indicating a noticeable loss of structural stiffness. The maximum vertical displacement reaches nearly 6 m.

Table 15. Variation of internal forces in various components after the failure of H₂-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	5390.7	1510.6	1370.4	1.11	0.03
N1a	5596.6	6140.4	1690.7	1600.7	1.10	0.02
N1b	1620.9	2120.2	678.6	394.2	1.31	0.18
N2a	3771.1	3760	1190.2	1130.2	1.00	0.02
N2b	5150.2	8680	2340.3	1460.5	1.69	0.17
T1a	2154.1	2570.2	800.8	539.8	1.19	0.12
T1b	2452.6	3130.4	951.3	598.4	1.28	0.14
T2a	4651.5	5230.5	1810.9	1270.3	1.12	0.12
T2b	5661.3	6900.9	2130.3	1690.7	1.22	0.08
B1	1143.2	3110.7	885.6	0.0	2.72	0.77
B2	4987.7	7040.3	1920.3	0.0	1.41	0.38
H1	2318.9	3680.1	1030.7	765.3	1.59	0.11
H2	10,668.1	14,100.2	5000.6	2030.2	1.32	0.28
V1a	−287.9	−600.4	−119.1	−100.8	2.09	0.06
V1b	−231.0	−439.3	−114.6	−63.1	1.90	0.22
V2a	−1113.5	−1390.1	−425.1	−270.4	1.25	0.14
V2b	−787.1	−3023.3	−994.0	−253.2	3.84	0.94

provides a statistical overview of the changes in internal forces. As indicated, the failure of H₂-1 affects various components, with a more pronounced impact on B₁, V_1a, and V_2b. The internal force dynamic coefficient β₁ for the V_2b strut in the outer ring reaches 3.84, and the internal force variation coefficient β₂ for internal force changes reaches 0.94. From Figure 16c, it is evident that the two lower chord nodes connected to H₂-1, located within the hexagonal honeycomb grid of the outer ring, experience noticeable deformation.

3.14. Failure of Component V_1a-1

V_1a is an internal ring compression brace that connects the upper chord class-a nodes with the lower chord nodes. Following the rupture, Figure 17a,b illustrates the time-dependent variations in internal forces and vertical displacements of V_1a and its associated nodes. After approximately 4 s of vibration, the internal forces in the brace stabilize, prompting a redistribution of forces within the structure and a slight increase in the internal forces of each V_1a brace. Notably, the upper chord node 1_a, which is linked to V_1a-1, experiences a significant downward displacement, while the remaining nodes exhibit negligible changes. Additional details regarding the changes in internal forces for other components can be found in the provided

Table 16. Variation of internal forces in various components after the failure of V_1a-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4949.7	4903.4	4848.6	1.02	0.01
N1a	5596.6	5689.7	5630.6	5591.3	1.02	0.01
N1b	1620.9	1656.5	1634.3	1608.5	1.02	0.02
N2a	3771.1	3788.1	3770.5	3761.7	1.00	0.01
N2b	5150.2	5206.8	5150.5	5138.5	1.01	0.01
T1a	2154.1	2215.2	2191.0	2152.8	1.03	0.02
T1b	2452.6	2510.3	2455.9	2441.7	1.02	0.01
T2a	4651.5	4676.6	4655.9	4633.4	1.01	0.01
T2b	5661.3	5686.2	5662.7	5644.7	1.00	0.01
B1	1143.2	1175.4	1145.1	1137.8	1.03	0.01
B2	4987.7	5091.8	5004.8	4941.4	1.02	0.01
H1	2318.9	2379.1	2315.0	2264.5	1.03	0.02
H2	10,668	10,755	10,667.0	10,649.4	1.01	0.01
V1a	−287.9	−320.3	−299.3	−288.9	1.11	0.04
V1b	−231.0	−332.1	−311.5	−231.8	1.44	0.34
V2a	−1113.5	−1128.1	−1115.4	−1091.2	1.01	0.02
V2b	−787.1	−801.0	−790.7	−780.4	1.02	0.01

. The rupture of V_1a-1 predominantly impacts the internal forces of V_1b, with the dynamic internal force coefficient β₁ reaching 1.44 and the internal force variation coefficient β₂ reaching 0.34. Upon inspecting Figure 17c, localized collapse becomes evident in the inner ring’s hexagonal grid, where support is lost.

3.15. Failure of Component V_1b-1

V_1b-1 represents a type-b compression strut in the inner ring. After its failure, the internal forces and vertical displacements of each node, as depicted in Figure 18a,b, undergo distinct changes. Following a prominent oscillation, the internal forces of V_1b stabilize, leading to a redistribution of forces within the structure. With the exception of V_1b-3, which experiences a significant increase in internal forces, the remaining V_1b elements undergo a slight reduction in internal forces. Notably, the vertical displacement of node 1_b exhibits substantial fluctuations, reaching 0.46 m within the first second before settling at around 0.3 m. Conversely, the displacements of other nodes remain relatively consistent. The accompanying

Table 17. Variation of internal forces in various components after the failure of V_1b-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4916.4	4871.0	4845.9	1.01	0.01
N1a	5596.6	5653.0	5607.1	5569.2	1.01	0.01
N1b	1620.9	1706.3	1675.3	1612.9	1.05	0.04
N2a	3771.1	3806.6	3791.6	3764.8	1.01	0.01
N2b	5150.2	5200.8	5165.0	5127.4	1.01	0.01
T1a	2154.1	2194.3	2169.3	2080.1	1.02	0.04
T1b	2452.6	2549.5	2508.4	2437.9	1.04	0.03
T2a	4651.5	4704.9	4679.4	4611.1	1.01	0.01
T2b	5661.3	5720.0	5680.4	5626.9	1.01	0.01
B1	1143.2	1204.9	1174.2	1025.5	1.05	0.13
B2	4987.7	5054.4	4993.1	4954.0	1.01	0.01
H1	2318.9	2349.0	2314.9	2265.4	1.01	0.02
H2	10,668.1	10,725.5	10,665.8	10,641.9	1.01	0.00
V1a	−287.9	−428.2	−392.9	−289.4	1.49	0.36
V1b	−231.0	−250.0	−241.5	−234.0	1.08	0.03
V2a	−1113.5	−1196.4	−1136.7	−1096.7	1.07	0.04
V2b	−787.1	−807.1	−800.8	−769.8	1.03	0.04

provides a statistical overview of internal force variations in other components, revealing no discernible impact on any particular element. Upon examining Figure 18c, it becomes evident that the hexagonal honeycomb grid in the inner ring of the first module undergoes only minor deformations, avoiding structural collapse.

3.16. Failure of Component V_2a-1

After the rupture of V_2a-1, the internal forces and vertical displacements of each node in V_2a undergo changes over time, as depicted in the accompanying Figure 19a,b. Following an initial significant fluctuation, the internal forces in V_2a-2 increase, while the remaining similar bracing members experience a minor reduction in internal forces. The vertical displacements of the nodes are predominantly affected by the failure of V_2a-1, particularly impacting node 1_b in the inner ring and node 2_a in the outer ring. Notably, during the fluctuation process, node 2_a exhibits a maximum displacement of 1.1 m. The statistical analysis of internal force variations for other components is provided in

Table 18. Variation of internal forces in various components after the failure of V_2a-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	5534.3	4827.3	4629.1	1.14	0.04
N1a	5596.6	6310.9	5512.8	5463.3	1.13	0.01
N1b	1620.9	2064.1	1838.8	1513.7	1.27	0.20
N2a	3771.1	4138.6	3728.2	3676.9	1.10	0.01
N2b	5150.2	5599.3	5129.3	4766.8	1.09	0.07
T1a	2154.1	2765.4	2440.0	1971.0	1.28	0.22
T1b	2452.6	3144.3	2840.3	2257.9	1.28	0.24
T2a	4651.5	5180.2	4704.9	4508.7	1.11	0.04
T2b	5661.3	6207.8	5687.2	5176.0	1.10	0.09
B1	1143.2	1690.0	1410.0	439.0	1.48	0.85
B2	4987.7	5750.0	5030.0	4300.0	1.15	0.15
H1	2318.9	2820.0	2460.0	2270.0	1.22	0.08
H2	10,668.1	12,000.0	10,600.0	9980.0	1.12	0.06
V1a	−287.9	−347.0	−291.0	−264.0	1.21	0.09
V1b	−231.0	−437.0	−272.0	−176.0	1.89	0.42
V2a	−1113.5	−1390.0	−1250.0	−1070.0	1.25	0.16
V2b	−787.1	−1400.0	−1120.0	−772.0	1.78	0.44

. The failure of V_2a-1 has a relatively even impact on other types of components, with V_1b being significantly influenced, as evidenced by its internal force dynamic coefficient β₁ of 1.89 and internal force variation coefficient β₂ of 0.42. Upon observing Figure 19c, localized collapse becomes evident in the vicinity of node 2_a within the first substructure.

3.17. Failure of Component V_2b-1

V_2b-1 is a b-class compression strut in the outer ring. After its rupture, the internal forces and vertical displacements of each node in V_2b vary over time, as shown in Figure 20a,b. Within the first 4 s following the rupture, all V_2b internal forces undergo significant fluctuations. However, they gradually stabilize afterward, with only V_2b-2 experiencing a slight increase in internal force. The remaining V_2b internal forces exhibit no notable changes. Similarly, the displacements of the nodes follow a similar pattern, with node 2_b showing a distinct downward displacement of up to approximately 0.7 m. Once the structure regains stability, the displacement decreases to around 0.45 m. The statistical analysis of internal force variations for other components can be found in

Table 19. Variation of internal forces in various components after the failure of V_2b-1.

Components	F (kN)	Fdmax (kN)	Fmax (kN)	Fmin (kN)	β1	β2
N1	4855.9	4916.4	4830.0	4802.9	1.01	0.01
N1a	5596.6	5661.5	5559.3	5529.5	1.01	0.01
N1b	1620.9	1677.8	1638.2	1600.3	1.04	0.02
N2a	3771.1	3934.1	3830.7	3682.6	1.04	0.04
N2b	5150.2	5740.3	5447.9	5107.4	1.11	0.07
T1a	2154.1	2195.1	2149.4	2087.5	1.02	0.03
T1b	2452.6	2552.1	2469.8	2384.6	1.04	0.03
T2a	4651.5	4740.3	4659.3	4399.3	1.02	0.06
T2b	5661.3	6021.7	5734.3	5606.1	1.06	0.02
B1	1143.2	1208.8	1162.3	963.3	1.06	0.17
B2	4987.7	5226.9	5029.3	4230.1	1.05	0.16
H1	2318.9	2407.3	2325.0	2299.0	1.04	0.01
H2	10,668.1	10,914.7	10,619.4	10,378.5	1.02	0.02
V1a	−287.9	−319.8	−292.3	−274.8	1.11	0.06
V1b	−231.0	−242.0	−229.2	−203.2	1.05	0.11
V2a	−1113.5	−1374.7	−1266.4	−1111.0	1.23	0.14
V2b	−787.1	−826.0	−797.1	−787.4	1.05	0.01

, indicating that the impact of V_2b-1 failure on other component types is relatively uniform, with no significantly affected components. Figure 20c also serves as evidence that the structure undergoes localized deformation without collapsing.

4. Components Importance Level Evaluation

Table 20. Comparison of structural performance after the failure of each component and the importance level.

Failure Components	Number of Affected Significant Components	Number of Relaxation Components	UZmax (m)	Localized Damage	Importance Level
N1	4	0	1.5	√	Medium
N1a	4	2	5.2	√	High
N1b	2	0	1.9	√	Medium
N2a	3	2	−4	√	High
N2b	4	2	4.5	√	High
T1a	4	0	1.9	√	Medium
T1b	4	0	0.6	√	Medium
T2a	6	4	−3	√	High
T2b	4	4	2.5	√	High
B1	0	0	−0.41	×	Low
B2	0	0	−0.13	×	Low
H1	4	1	−2.2	√	High
H2	6	2	−3.5	√	High
V1a	0	0	−0.17	×	Low
V1b	0	0	−0.31	×	Low
V2a	2	0	−0.8	√	Medium
V2b	0	0	−0.42	×	Low

compares various indicators after the failure of different components and assesses the importance of the components based on the rules outlined in Section 2.4.

The failure of the upper chord ridge cables N₁, N_1b, T_1a, or T_1b resulted in significant effects on the remaining components and localized damage. However, no relaxation components were observed within the structure. Therefore, according to the rules, the importance level of these components can be classified as “Medium”. On the other hand, after the failure of the upper chord components N_1a, N_2a, N_2b, T_2a, or T_2b, the remaining components were significantly affected, leading to localized damage to the structure. Simultaneously, the presence of relaxation components within the structure was observed. Consequently, the importance level of these components should be categorized as “High”.

After the failure of diagonal cables B₁ or B₂, no significant effects or relaxation components were observed in the structure, and the vertical displacement of the structure did not exceed the limit. Therefore, the importance level of both types of diagonal cable can be classified as “Low”. As ring cables with higher internal forces in the structure, they primarily contribute to the structural stiffness and usually have a significant impact on the structure. After the failure of H₁ or H₂, it results in significant effects on the remaining components, leading to localized damage, and at the same time, the presence of relaxation components is observed within the structure. Consequently, the importance level of both types of tension cables is categorized as “High”.

The struts are located in the lower chord of the structure and are the only compression components in the structure. After the failure of struts V_1a, V_1b, or V_2b, no significant effects or relaxation components were observed in the structure, and the vertical displacement of the structure did not exceed the limit. Therefore, the importance level of these struts can be classified as “Low”. The situation is quite different after the failure of the strut V_2a. It not only significantly affects the internal forces of some remaining components but also leads to localized damage. According to the rules, the importance level of strut V_2a should be classified as “Medium”.

5. Conclusions

In this study, the dynamic response and failure patterns of the drum-shaped honeycomb type III cable dome with quad-strut layout were investigated when experiencing localized failure of individual components due to accidental events or severe overloading by establishing a finite element numerical model and utilizing nonlinear dynamic analysis methods. Based on the computational results of single-component failures, the importance levels of different types of components within the overall structure were determined.

In conclusion, this research contributes to the understanding of the stability of cable dome structures with this configuration. The findings offer valuable insights for structural design and safety assessment, making this research a valuable addition to the field of structural engineering. The main conclusions drawn in this paper are as follows:

• The ridge cables play a critical role in the structure, and their failure can lead to different levels of localized collapse. Moreover, certain ridge cables are prone to relaxation. The circumferential ridge cables N hold slightly greater importance compared to the transverse ridge cables T.
• When a single inner or outer diagonal cable B fails, it does not cause substantial deformation of the structure. Consequently, the importance of the diagonal cables is relatively low compared to other components.
• The annular cable H holds significant importance in the structure due to its substantial internal forces and its crucial contribution to the overall stiffness. It plays a vital role in maintaining the equilibrium of internal forces within the structure. The failure of the annular cable has a pronounced impact on the internal forces of other components and the displacement of nodes, ultimately resulting in localized structural collapse.
• The failure of strut V leads to a partial loss of vertical stiffness in the structure, resulting in a significant decrease in its load-bearing capacity. Since the type-a struts are vertically arranged, they contribute significantly to the overall vertical stiffness. As a result, the importance of type-a struts is markedly higher than that of type-b struts. Based on the computational results, it appears that outer ring struts are more important than inner ring struts.
• In the case of failure of a single component in this particular cable dome structure, no overall collapse occurs. Analyzing the dynamic response reveals that the outer ring significantly contributes to the overall stiffness of the structure. Consequently, the safety rating of the components in the outer ring is consistently higher than that of components in the inner ring.
• In practical engineering design, for components with medium or high importance levels, more conservative design methods should be adopted to ensure the stability and safety of the structure.